Theorem nnssn0s 28327

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28322	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4135	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 4029	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3947   ⊆ wss 3950  {csn 4625   0_s c0s 27868  ℕ_0scnn0s 28319  ℕ_scnns 28320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953  df-ss 3967  df-nns 28322

This theorem is referenced by:  nnssno  28328  nnn0s  28333  nnn0sd  28334  nnsgt0  28343